WWOORRKKIINNGG PPAAPPEERR NNOO.. 228888

Transparency and Product Differentiation

with Competing Vertical Hierarchies

Matteo Bassi, Marco Pagnozzi and Salvatore Piccolo

July 2011

University of Naples Federico II

University of Salerno

Bocconi University, Milan

CSEF - Centre for Studies in Economics and Finance DEPARTMENT OF ECONOMICS – UNIVERSITY OF NAPLES

80126 NAPLES - ITALY Tel. and fax +39 081 675372 – e-mail: csef@unisa.it

WWOORRKKIINNGG PPAAPPEERR NNOO.. 228888

Transparency and Product Differentiation

with Competing Vertical Hierarchies

Matteo Bassi*, Marco Pagnozzi** and Salvatore Piccolo***

Abstract We revisit the choice of product differentiation by competing firms in the Hotelling model, under the assumption that firms are vertically separated, and that retailers choose products’ characteristics. We show that retailers with private information about their marginal costs choose to produce less differentiated products than retailers with no private information, in order to increase their information rents. Hence, information asymmetry increases social welfare because it induces firms to sell products that appeal to a larger number of consumers. The socially optimal level of transparency between manufacturers and retailers depends on the weight assigned to consumers’ surplus and trades of two effects: higher transparency reduces price distortion but induces retailers to produce excessively similar products. Keywords: product differentiation, vertical relations, transparency, regulation. JEL Classification: D43, D82, L13, L51. Acknowledgements: We would like to thank Giacomo Calzolari, Riccardo Martina and Tommaso Valletti for extremely helpful comments.

* University of Salerno and CSEF. E-mail: matteo.bassi@unina.it ** Università di Napoli Federico II and CSEF. E-mail: pagnozzi@unina.it. *** Università di Napoli Federico II and CSEF. E-mail: salvapiccolo@gmail.com.

Table of contents

1. Introduction

2. The Model

3. Full Transparency Benchmark

4. Equilibrium with Asymmetric Information

5. Optimal Transparency

6. Conclusions

Appendix

References

1. Introduction

We consider two rms that sell substitute products and choose the degree of horizontaldi¤er-

entiation among their products. Following the classic Hotelling (1929) model, we assume that

rms choose where to locate on a line; consumers are distributed along the line and pay a trans-

portation cost to reach a rm and purchase its product. Alternatively, each point on the line

may be interpreted as a possible variety of a product (e.g., a di¤erent amount of some product

characteristic). The point at which each consumer is located denotes its most preferred variety

while a rms location represents the variety that it chooses to produce. Transportation cost

can be seen as the loss of utility of a consumer that purchases a variety that is di¤erent from

its most preferred one. This model captures the idea that di¤erent consumers prefer di¤erent

varieties of a product, and rms can choose the degree of di¤erentiation among their products

by choosing their specic characteristics.

According to the principle of di¤erentiation, rms want to di¤erentiate their products in

order to soften price competition. In various contexts, rms want to maximize di¤erentiation

and locate as far as possible from their rivals (e.g., with quadratic transportation costs see

DAspremont et al., 1979). This is consistent with the observation that rms often search for

market niches to position their products with respect to competitorsproducts. However, there

are a number of factors that limit rmsincentive to di¤erentiate their products: concentration

of demand on a particular variety of the product (that induces rms to locate where the demand

is see Economides, 1986, and De Palma et al., 1985), xed costs of production (that limit

the number of di¤erent varieties that rms are willing to produce), positive externalities in

production or demand among rms producing similar products, etc.

We introduce a novel reason that may induce vertically separated rms to produce less dif-

ferentiated products: asymmetric information between manufacturers and retailers.1 Retailers

often undertake non-market activities such as product design, advertising campaigns, and qual-

ity enhancing investments, that determine the degree of product di¤erentiation with respect to

competitors. But retailers who have private information about their marginal costs of produc-

tion also have an incentive to choose product characteristics that appeal to a larger number

of consumers, in order to increase sales and, hence, their information rent. Therefore, when

retailers choose products characteristics before prices, asymmetric information induces them

to choose less di¤erentiated products (than without private information), even though this in-

creases competition. The availability of less di¤erentiated products increases consumerswelfare

because it reduces transportation costs (which can be interpreted as more consumers acquiring

a product that is relatively similar to their most preferred ones).

Building on this result, we consider a regulator who can control the level of asymmetric

information between manufacturers and retailers (i.e., the dispersion of retailers private infor-

1There is a large empirical literature showing that asymmetric information a¤ects pricing decision in industrieswhere rms are vertically separated (see, e.g., Lafontaine and Slade, 1997, and Lanfontaine and Shaw, 1999).

2

mation), which we interpret as transparency. In practice, rmsaccounting rules are subject to

regulation that, directly or indirectly, determines the quality and the quantity of information

that rms need to report to the public. For instance, when they are subject to stricter account-

ing standards, retailers must provide more detailed accounting reports, which can be observed

by their suppliers, whereby reducing retailersprivate information.

It is often argued that a regulator who aims to maximize social welfare should always reduce

asymmetric information between manufacturers and retailers, because asymmetric information

induces rms to choose ine¢ ciently high prices.2 By contrast, we prove that, when a regulator

can control the dispersion of retailers private information, he never chooses to impose full

transparency i.e., to eliminate private information altogether although this would eliminate

the distortion in retail prices due to private information. The reason is that, without asymmetric

information, rms maximize product di¤erentiation, thus reducing welfare, while retailers with

private information produce less di¤erentiated products.

The socially optimal level of transparency depends on the weight assigned to consumerssur-

plus. Reducing transparency has two opposite e¤ects on social welfare. On the one hand, lower

transparency tends to reduce welfare because it induces manufacturers to charge ine¢ ciently

high prices: a price distortion e¤ect. On the other hand, lower transparency induces retailers

to produce products that are relatively more di¤erentiated (but not maximally di¤erentiated),

thus reducing consumerstransportation costs and increasing welfare: a product di¤erentiation

e¤ect.3 When the regulator assigns a low weight to consumerssurplus, he chooses the maximal

degree of asymmetric information compatible with the market being fully covered. In this case,

the price distortion e¤ect is weaker than the product di¤erentiation e¤ect. By contrast, when

the weight assigned to consumerssurplus takes intermediate values, the optimal degree of asym-

metric information is positive, but not maximal, so as to balance the price distortion and the

product di¤erentiation e¤ects. Finally, when the regulator assigns a high weight to consumers

surplus, he minimizes asymmetric information because the price distortion e¤ect prevails.

On the normative ground, our analysis o¤ers a justication for regulatory policies that

allow for lower (or imperfect) standards of transparency.4 Although we analyze a stylized IO

model, our results are more general and may be applied to procurement contracting, executive

compensations, patent licensing or credit relationships, when there are competing hierarchies

and non-contractable product di¤erentiation activities, such as investments in product design,

advertising campaigns, R&D etc.

The rest of the paper is organized as follows. Section 2 presents the model. After discussing

2The literature on supply chains, for instance, shows how the presence of privately informed retailers leads toequilibrium prices higher than marginal costs see Blair and Lewis (1994), Gal-Or (1991, 1999), Khun (1997),Martimort (1996), and Martimort and Piccolo (2007).

3When transparency increases, retailers choose products that are too similar from a social welfares point ofview, because their characteristics are most preferred by fewer potential consumers.

4Bennardo et al. (2010), Calzolari and Pavan (2006), Taylor (2004), and Maier and Ottaviani (2009) analyzetransparency in vertical contracting. In contrast to our paper, this literature does not consider competing verticalhierarchies.

3

the benchmark case of full transparency in Section 3, in Section 4 we consider the case of

asymmetric information. Section 5 analyzes the choice of transparency by a regulator who

maximizes social welfare. Finally, Section 6 concludes. All proofs are in the appendix.

2. The Model

Players and environment. We consider the linear city model introduced by Hotelling(1929). There is a unit mass of consumers uniformly distributed with density 1 over [0; 1]. Two

vertical structures, each composed by one manufacturer and one retailer, produce a homogeneous

good and are located at a1 and (1� a2), respectively, where without loss of generality a1+a2 � 1(a1 = a2 = 0 implies maximal di¤erentiation). Specically, manufacturerM1 and retailer R1 are

located at a1, while manufacturer M2 and retailer R2 are located at (1� a2). The location of avertical structure is chosen by the retailer. We assume that each vertical structure can choose

only one location i.e., can produce a single variety of the good because of xed costs.5

Each consumer has a valuation v for a single unit of the good. For simplicity, we assume

that v ! +1, so that each consumer always buys one unit, regardless of the price. Consumerspay a quadratic transportation cost to reach the vertical structures. Specically, a consumer

located at x 2 [0; 1] pays t (x� a1)2 to buy from R1 and t (1� a2 � x)2 to buy from R2.Given the retail prices of the goods produced by the two vertical structures, p1 and p2, a

consumer located at x buys from R1 if and only if

p1 + t (x� a1)2 < p2 + t (1� a2 � x)2 :

Therefore, in an interior solution, the demand for the good sold by Ri is

Di (pi; pj) =1 + ai � aj

2+

pj � pi2t (1� ai � aj)

; i; j = 1; 2; i 6= j:

Before being o¤ered a contract from his manufacturer, Ri privately observes his constant

marginal cost of production �i, which is distributed uniformly on � � [�� �; �+ �], so that itsc.d.f. is F (�i) =

�i�(���)2� with mean � and variance

�2

3 > 0. We assume that: (i) � 2 f0g[[�; �];(ii) � > 0 and � ' 0; (iii) � < t4 and � '

t4 .6 When � = 0 there is full transparency and

retailersmarginal costs are equal to �. When � 6= 0 there is asymmetric information betweenmanufacturers and retailers. We also assume that � < �, so that marginal costs are always

positive.

Contracts. Contracts between manufacturers and retailers are private i.e., they cannot be5Matsushima and Matsumura (2003) analyze an Hotelling model with two vertically integrated rms and cost

uncertainty.6The assumption on � ensures the existence of a solution to the regulators problem in Section 5 (see the proof

of Proposition 2). The assumption on � ensures that retailersmarginal costs cannot di¤er too much, so that ina symmetric equilibrium each retailers demand is always strictly positive (see the proof of Proposition 1).

4

observed by competitors. We consider resale price maintenance (RPM) contracts and use

the Revelation Principle to characterize the equilibrium of the game. Hence, Mi o¤ers a menu

(pi(�̂i); Ti(�̂i))�̂i2� to Ri, where given Ris report �̂i to Mi, pi(�̂i) represents the retail price at

which Ri has to sell to nal consumers and Ti(�̂i) is the franchise fee paid by Ri to Mi. In the

appendix we prove that, with private contracts, our model with RPM contracts is equivalent to

a model in which each manufacturer o¤ers menus of two-part tari¤s composed by a wholesale

price and a franchise fee, contingent on the retailers report.

Timing. The timing of the game is as follows:

1. Retailers simultaneously and independently choose their locations: a1 and a2. Locations

are publicly observable.

2. Retailers privately observe their costs.

3. Manufacturers o¤er contracts and retailers choose whether to accept them. If Ri accepts

Mis o¤er, then he reports his type and pays the franchise fee.

4. Retailers announce retail prices and the market clears.

Equilibrium concept. The solution concept is Perfect Bayesian Equilibrium (PBE). Sincecontracts are private, we have to make an assumption on retailersbeliefs about their competi-

torsbehavior. We assume that, regardless of the contract o¤ered by his own manufacturer, a

retailer always believes that the other manufacturer o¤ers the equilibrium contract,7 and that

each retailer expects the rival retailer to truthfully report his type to the manufacturer in a

separating equilibrium.

3. Full Transparency Benchmark

Suppose rst that there is full transparency (� = 0). Hence, information is complete since

retailerscosts are deterministic and equal to �, and Mi o¤ers a contract (pi; Ti) that extracts

the whole surplus from her retailer i.e., she charges

Ti = Di(pi; pj)(pi � �);

where pj is the equilibrium price chosen by Mj . Given retailerschoice of location, Mi chooses

the retail price to maximize prot i.e., to solve

maxpiDi(pi; p

�j ) (pi � �) :

7See Pagnozzi and Piccolo (2010) for an analysis of the roles of beliefs when contracts between manufacturersand retailers are private.

5

Since retailers obtain no rent when they have no private information, they are indi¤erent among

any location choice. Following a standard convention, we assume that, when indi¤erent, retailers

choose the location that maximizes manufacturersprot.

Lemma 1. In a symmetric equilibrium rms locate at 0 and 1 i.e., they choose maximaldi¤erentiation and manufacturers choose a retail price equal to t+ �.

The intuition for this result is straightforward. By producing a di¤erentiated product, a

manufacturer reduces price competition and enjoys some market power over consumers distrib-

uted around her location. This is the principle of di¤erentiation. Of course, rmsprices are

increasing in t, because when the transportation cost is higher products are perceived as more

di¤erentiated by consumers, and hence manufacturers compete less ercely to attract consumers

by charging lower prices. Prices are also increasing in �, because when marginal costs are higher,

manufacturers choose higher prices in equilibrium.

4. Equilibrium with Asymmetric Information

Suppose that � 6= 0. We focus on a symmetric separating equilibrium. Let p�i (�i), i = 1; 2, bethe retail price chosen by Mi when Ris cost is �i, given the locations chosen by retailers. In

a separating equilibrium, a manufacturers contract must be incentive feasible i.e., it must

satisfy the retailers participation and incentive compatibility constraints.

Given an incentive compatible menu (pi(:); Ti(:)), Ris information rent is

Ui (�i) = (pi(�i)� �i)Z �+����

Di(pi (�i) ; p�j (�j))dF (�j)� Ti(�i); i = 1; 2:

Incentive compatibility implies that

Ui(�i) = max�̂i2�

�(pi(�̂i)� �i)

Z �+����

Di(pi(�̂i); p�j (�j))dF (�j)� Ti(�̂i)

�;

which yields the rst- and second-order local conditions for incentive compatibility

_Ui (�i) = �Z �+����

Di(pi(�i); p�j (�j))dF (�j) ; (4.1)

and

�@R �+���� D

i(pi(�i); p�j (�j))dF (�j)

@pi_pi (�i) � 0 ) _pi (�i) � 0: (4.2)

Conditions (4.1) and (4.2), together with the participation constraint

Ui (�i) � 0; 8�i 2 �; (4.3)

6

dene the set of incentive-feasible allocations for Mi and Ri. Therefore, Mi solves the following

optimization program

maxfpi(:);Ui(:)g

Z �+����

Z �+����

�Di(pi(�i); p

�j (�j)) (pi(�i)� �i)� Ui (�i)

dF (�j) dF (�i) ;

subject to conditions (4.1), (4.2) and (4.3).

To solve this program, we rst assume that _pi (�i) � 0, and then check that this conditionholds ex post. It follows that Ui (�i) is decreasing and the participation constraint is binding

when �i = �+ �. Hence, Ris information rent is

Ui (�i) =

Z �+��i

Z �+����

Di(pi(x); p�j (�j))dF (�j)dx: (4.4)

This rent is increasing in consumersdemand for the good sold by Ri. The reason is that a

retailer with a low marginal cost obtains a higher utility by mimicking retailers with higher

marginal costs when those retailers sell a higher quantity on average i.e., the information rent

of a type is increasing in the quantity sold by less e¢ cient types. Since the demand for the good

sold by Ri is increasing in ai (because locating closer to the center attracts more customers), this

provides an incentive for a retailer to produce a product that is more similar to his competitors

product.

Using expression (4.4), integrating by parts, and substituting inMis objective function yield

the simplied program

maxpi(:)

Z �+����

Z �+����

�Di(pi(�i); p

�j (�j))

�pi(�i)� �i �

F (�i)

f (�i)

��dF (�j) dF (�i) :

The rst-order condition for this program is

p�i (�i) = �i +F (�i)

f (�i)�

R �+���� D

i(p�i (�i); p�j (�j))dF (�j)R �+�

���@@piDi(p�i (�i); p

�j (�j))dF (�j)

: (4.5)

Lemma 2. Given retailerslocations a1 and a2, Mi chooses the retail price

p�i (�i) = �i + � +t3 (1� ai � aj) (3� ai + aj) ; i = 1; 2: (4.6)

By equation (4.5), the retail price distortion with respect to marginal cost is increasing in the

hazard rate F (�i)f(�i) . The reason is that, at a best-response to the price schedule p�j (�j),Mi chooses

pi so as to equalize her virtual marginal revenue to zero. Under asymmetric information, the cost

parameter �i is replaced by a higher virtual cost parameter �i+F (�i)f(�i)

, so that the allocation of a

high-cost type becomes less attractive to a low-cost type, and the latters incentive to misreport

his marginal cost is mitigated. In other words, increasing the price of an agent with type �jreduces the information rents of agents with types lower than �j .

7

Consider now retailerschoice of locations. Before observing his marginal cost, in order to

maximize his expected information rent, Ri solves

maxai

Z �+����

Z �+��i

Z �+����

Di(p�i (x); p�j (�j))dF (�j)dxdF (�i) :

Two opposite e¤ects determine retailerschoice. On the one hand, if a retailer reduces the degree

of di¤erentiation of his product vis-à-vis his rival, the rival rm reacts by reducing his own price:

price competition is more intense with less di¤erentiated products. This (standard) strategic

e¤ect tends to reduce demand and, hence, the retailers information rent. On the other hand,

however, choosing a location closer to the center allows a retailer to attract more customers,

since they have to pay a lower transportation cost to acquire the retailers product. This sales

e¤ect tends to increase information rent.

Proposition 1. In a symmetric PBE, both retailers choose

a� (�) = 12 �12

q�t ;

and set a retail price equal to

p�(�i) = �i + � +pt�:

Compared to the complete information benchmark, where the principle of di¤erentiation

applies, retailers choose a lower degree of di¤erentiation among their products, in order to

increase their information rents. Retailers would jointly prefer to choose maximal di¤erentiation

i.e., a1 = a2 = 0 since these are the symmetric locations that maximize total retailers

rents. These are the locations chosen by retailers when � = 0. However, when � 6= 0, if oneretailer chooses ai ' 0, the other retailer has an incentive to locate closer to the center in orderto increase his information rent. In other words, when product are very di¤erentiated, the sales

e¤ect dominates. Anticipating this, the rst retailer moves closer to the center as well: the

choices of locations are strategic complements for retailers. Hence, retailers face a prisoners

dilemmawhen choosing their locations.

Notice, however, that a� is decreasing in �, for � 2 [�; �]. A higher � implies a higher retailprice, because more private information creates more price distortion, and a lower a� i.e., more

product di¤erentiation. The reason is that, when � is high, retailers expect retail price to be

higher and, hence, have a lower incentive to produce less di¤erentiated product to increase sales.

So retailers can produce more di¤erentiated products, which increase prot for the strategic

e¤ect described above. As � ! 0, a� ! 12 : when asymmetric information vanishes, retailerstend to eliminate product di¤erentiation altogether, since they have the strongest incentive to

increase sales in order to obtain an information rent. Figure 4.1 summarizes rmschoices of

product di¤erentiation as a function of �.

A higher t implies a higher a� i.e., less di¤erentiation and a higher retail price, because

8

a*

σσ00

0.5

σ

0.25

( )σ

Figure 4.1: Retailerschoice of location

when the transportation cost is high, consumers perceive products as more di¤erentiated, thus

reducing price competition among rms.

5. Optimal Transparency

In this section, we analyze the choice of the level of transparency between manufacturers and

retailers, by a regulator who is interested in maximizing expected welfare. Specically, we assume

that the regulator maximizes a weighted sum of consumer surplus and rmsprots (i.e., the

sum of manufacturersprots and retailersinformation rents). Given marginal costs �1 and �2,

retail prices p1 and p2, and locations a1 and a2, the welfare function is

W (:) � �"v �

Z D1(p1;p2)0

�p1 + t (x� a1)2

�dx�

Z 1D1(p1;p2)

�p2 + t (1� a2 � x)2

�dx

#+

(1� �)�D1(p1; p2)(p1 � �1) +

�1�D1(p1; p2)

�(p2 � �2)

�

= �v �D1(p1; p2) [(2�� 1)p1 + (1� �)�1]� �Z D1(p1;p2)0

t (x� a1)2 dx+

��1�D1(p1; p2)

�[(2�� 1) p2 + (1� �)�2]� �

Z 1D1(p1;p2)

t (1� a2 � x)2 dx;

9

where � 2 [12 ; 1] is the weight assigned to consumerssurplus. When � =12 , the regulator treats

consumers and rms symmetrically and simply minimizes transportation and production costs

(since in our model the total demand is xed). When � = 1, the regulator maximizes consumers

surplus and, hence, minimizes retail prices and transportation costs. Ceteris paribus, a higher

� implies that the regulator is relatively more concerned about reducing retail prices.

We assume that the regulator can choose � to determine the level of transparency i.e.,

the amount of retailersprivate information with respect to manufacturers. By choosing � = 0,

the regulator imposes full transparency; by choosing � 2 [�; �], the regulator controls the levelof transparency when retailers have private information: increasing � reduces transparency

since it results in a higher amount of private information. For example, the regulator can

increase transparency by imposing more restricting accounting standards to retailers, that allow

manufacturers to infer more information about their retailersmarginal costs.

Hence, given the equilibrium choices of locations and retail prices, the regulator solves

max�2f0g[[�;�]

Z �+����

Z �+����

W (:) dF (�1) dF (�2) :

The analysis of Section 3 suggests that, with asymmetric information, the regulator can

prevent rms from choosing maximally di¤erentiated products.

Lemma 3. The regulator never chooses full transparency i.e., he always chooses � 6= 0.

Even if the regulator can completely eliminate asymmetric information between manufactur-

ers and retailers by imposing full transparency, he never chooses to do so. The reason is that,

with full transparency, retailers choose maximal product di¤erentiation while, with a positive

level of asymmetric information, they locate away from the extremes of the interval i.e.,

they produce less di¤erentiated products thus intensifying price competition and increasing

welfare.

In order to introduce the e¤ects on expected welfare of asymmetry between manufactur-

ers and private information by retailers, we rst consider the choice of a regulator who only

minimizes transportation costs or production costs.

Lemma 4. Assume that the regulator is only interested in minimizing transportation costs.Then the regulator chooses � = t� t

p32 and retailers locate (approximately) at 0:317 and 0:683.

Assume that the regulator is only interested in minimizing total production costs. Then the

regulator chooses � = � and retailers locate (approximately) at 14 and34 .

With symmetric rms and no uncertainty about marginal costs, the locations that minimize

transportation costs are 14 and34 . With asymmetric rms, however, the regulator induces re-

tailers to locate closer to the center to minimize transportation costs, since otherwise contested

consumers (located between the two rms) may be forced to pay very high transportation costs

10

in order to purchase from the most e¢ cient rm. By contrast, if the regulator wants to reduce

production costs, he induces rms to locate further away from each other in order to increase

the number of contested consumers who purchase from the most e¢ cient rm.

We now consider the optimal choice of transparency by the regulator.

Proposition 2. There exist � and �, with � > � > 12 , such that:

� For � 2 [12 ; �), the regulator chooses �. Retailers locate approximately at14 and

34 .

� For � 2 [�; �], the regulator chooses � (�; t), where � < � (�; t) < �, @�(�;t)@� < 0 and@�(�;t)@t > 0. Retailers choose a

� (� (�; t)), where 14 < a� (� (�; t)) < 12 and

@a�(�(�;t))@� > 0.

� For � 2 (�; 1], the regulator chooses �. Retailers locate approximately at 12 .

Reducing the dispersion of retailersprivate information has two opposite e¤ects on social

welfare. On the one hand, asymmetric information reduces welfare because it induces manu-

facturers to distort prices upward in order to minimize retailers information rents a price

distortion e¤ect. On the other hand, asymmetric information a¤ects retailerschoice of prod-

uct di¤erentiation: minimizing asymmetric information induces retailer to locate too close to

the center (thus producing products that appeal to fewer consumers) a product di¤erentia-

tion e¤ect. The relative strength of these e¤ects depends on the weight assigned to consumers

surplus.

As � increases, the regulator chooses a higher level of transparency and induces retailers to

locate closer to the center i.e., to produce less di¤erentiated products. Hence, the optimal

level of transparency is decreasing in �. Notice, however, that the optimal level of transparency

is discontinuous since lim�!� � (�; t) > � ' 0. Figure 5.1 summarizes the regulators choice of� as a function of �.

When � is su¢ ciently high, the regulator assigns a high weight to consumerssurplus and,

hence, he minimizes � in order to reduce prices, even though this induces retailers to locate too

close to the center, compared to the location that minimizes transportation costs. In this case,

the price distortion e¤ect is stronger than the product di¤erentiation e¤ect. By contrast, when

� is su¢ ciently low, the regulator assigns a lower weight to consumerssurplus and, hence, he

maximizes � and reduces the level of transparency. In this case, the price distortion e¤ect is

weaker than the product di¤erentiation e¤ect. When � takes intermediate values, the regulator

chooses a strictly positive, but not maximal, degree of asymmetric information in order to

balance the price distortion and the product di¤erentiation e¤ects.

Finally, notice that Lemma 3 proves that the regulator never chooses full transparency, be-

cause there is always a non-zero level of transparency that yields higher social welfare. However,

the regulator may not be able to ne tune � and achieve the optimal level of transparency

described in Proposition 2. Nonetheless, the next lemma shows that any level of asymmetric

information (i.e., any level on transparency di¤erent form 0) generates higher social welfare than

full transparency.

11

λλ0.5λ

σ

σ

σ

1

Figure 5.1: Socially optimal transparency

Lemma 5. For any � 2 [12 ; 1], social welfare is always higher with asymmetric information (forany � 2 [�; �]) than with full transparency.

Therefore, even if the regulator is not fully able to select the desired level of transparency,

he always prefers that retailers have some private information. This o¤ers a justication for

regulatory policies that allow rms to have relatively low standards of transparency in their

accounting reports.

6. Conclusions

We have introduced a novel reason that may induce vertically separated rms to produce less

di¤erentiated products. When privately informed retailers undertake non-market activities that

determine the degree of product di¤erentiation with respect to competitors such as product

design, advertising campaigns, and quality enhancing investments they have an incentive to

choose product characteristics that appeal to a larger number of consumers, in order to increase

their information rent. This also increases consumerssurplus.

Market transparency a¤ects social welfare. When a regulator can control the degree of

asymmetric information between manufacturers and retailers, he never imposes full transparency,

because less strict transparency standards always produce higher welfare than full transparency.

Lower transparency may also be interpreted as retailers adopting riskier technologies, that entail

higher uncertainty about marginal costs. Therefore, in contrast to common wisdom, asymmetric

12

information may be socially benecial in our model.

13

A. Appendix

Proof of Lemma 1. The proof of this result is standard see, e.g., Tirole (1988), Ch. 7, p.281. �

Proof of Lemma 2. The rst-order necessary and su¢ cient condition for Mis program is

1 + ai � aj2

+

R �+���� p

�j (�j) dF (�j)� 2p�i (�i) + �i +

F (�i)f(�i)

2t (1� ai � aj)= 0

, p�i (�i) = 12 t (1� ai � aj) (1 + ai � aj) +12

Z �+����

p�j (�j) dF (�j) + �i � 12 (�� �) : (A.1)

Taking expectations with respect to �i,Z �+����

p�i (�i)dF (�i) =12 t (1� ai � aj) (1 + ai � aj) +

12

Z �+����

p�j (�j) dF (�j) +12 (�+ �) :

Hence,Z �+����

p�i (�i)dF (�i) = �+ � +23 t (1� ai � aj) (2ai + aj) + t (1� ai � aj)

2 ; i; j = 1; 2:

Substituting this equation in (A.1) yields equation (4.6). �

Proof of Proposition 1. Using the equilibrium prices in Lemma 2, Ris expected rent isZ �+����

Z �+��i

Z �+����

Di(p�i (x); p�j (�j))dF (�j)dxdF (�i) =

�(3� �t � 2ai � 4aj � a2i + a

2j )

6 (1� ai � aj):

Hence, Ris optimization program is

maxai

3� �t � 2ai � 4aj � a2i + a

2j

1� ai � aj;

yielding the rst-order necessary and su¢ cient condition

t(1 + a2i + a2j )� 2t(ai � aj + aiaj)� � = 0:

The relevant solution for ai is

ai(aj) = 1�q

�t + aj ;

which implies that ai(aj) is increasing in aj i.e., location choices are strategic complements.In a symmetric equilibrium, each retailer chooses ai = aj = a� = 12 �

12

p�t . Using equation

14

(4.6), the equilibrium retail price is

p�(�i) = �i + � + t (1� 2a�)= �i + � +

pt�: (A.2)

Since _p�(�i) > 0, the local second-order incentive compatibility constraint (4.2) is satised.Moreover, consider the global incentive compatibility constraint. Let the equilibrium franchisefee be

T �(�i) = (p�(�i)� �i)

Z �+����

Di(p� (�i) ; p� (�j))dF (�j)�

Z �+��i

Z �+����

Di(p�(x); p� (�j))dF (�j)dx:

The equilibrium contract must satisfy the following inequality

Ui (�i) � Ui��i; �

0� ; 8(�i; �0) 2 �2:,

Z �0�i

(_T �(x)�

@(p�(x)� �i)R �+���� D

i(p� (x) ; p� (�j))dF (�j)

@x

)dx � 0;

,Z �0�i

�_T �(x)� _p�(x)

�Z �+����

Di(p� (x) ; p� (�j))dF (�j)�p�(x)� �i2t (1� 2a�)

��dx � 0: (A.3)

Since _p�(�i) = 1 and the local rst-order incentive compatibility implies

_T �(�i) = �@(p�(�i)� �i)

R �+���� D

i(p� (�i) ; p� (�j))dF (�j)

@�i

= _p�(�i)

�Z �+����

Di(p� (�i) ; p� (�j))dF (�j)�

p�(�i)� �i2t (1� 2a�)

�; 8�i 2 �;

the left-hand side of (A.3) is Z �0�i

x� �i2t (1� 2a�)dx: (A.4)

Suppose, without loss of generality, that �0 > �i. Then x > �i and (A.4) is positive. Hence, theglobal incentive compatibility constraint holds at equilibrium.

Finally, we need to check that retailersdemand is (strictly) positive regardless of realizedcosts. Note that, since the equilibrium price is increasing with respect to marginal costs,

Di(p�(�+ �); p�(�� �)) > 0 ) Di(p�(�i); p�(�j)) > 0; 8(�i; �j) 2 �2:

Using the equilibrium price and the equilibrium localization choice,

Di(p�(�+ �); p�(�� �)) = 12 �q

�t :

Hence, our assumption that �t <14 ensures that demand is always positive. �

15

Proof of Lemma 3. In order to prove that the regulator never chooses full transparency, weshow that: (i) if � 6= 12 , welfare when � = � is strictly higher than when � = 0; (ii) if � =

12 ,

welfare when � = � is the same as when � = 0.First, when � = 0, retailers choose maximal di¤erentiation. Using the retail prices in Lemma

1, the welfare function is

W (:)j�=0 = �v � (2�� 1) (�+ t)� (1� �)�� �tZ 1

2

0x2dx� �t

Z 112

(1� x)2 dx

= �v + t� ���+ 2512 t

�:

Second, when � = � � 0, retailers locate at ae = 12 �12

q�t �

12 . Using the retail prices in

equation (A.2), R1s demand is

D1(p� (�1) ; p� (�2)) =

1

2+p� (�2)� p� (�1)2t (1� 2a�)

=1

2+

�2 � �12t (1� 2a�) :

Hence, the (expected) welfare function isZ �+����

Z �+����

W (:) 14�2d�1d�2

�����=�

=

= �v �Z �+����

Z �+����

(Z 12+

�2��12t(1�2a�)

0

h(2�� 1)p�(�1) + �t (x� a�)2 + (1� �)�1

idx+

�Z 112+

�2��12t(1�2a�)

h(2�� 1) p�(�2) + �t (1� a� � x)2 + (1� �)�2

idx

)14�2d�1d�2

� �v � ���+ t12

�� W (:)j�=0 :

The inequality is strict for � 6= 12 . �

Proof of Lemma 4. In order to minimize transportation costs, the regulator minimizes

Z �+����

Z �+����

"Z 12+

�2��12t(1�2a�)

0t (x� a�)2 dx+

Z 112+

�2��12t(1�2a�)

t (1� a� � x)2 dx#

14�2d�1d�2

= �6

q�t �

pt�4 +

�4 :

The rst-order necessary and su¢ cient condition is

14 +

14

q�t �

18

qt� = 0 , � = t�

tp32 :

16

Hence, retailers locate at 34 �p34 .

In order to minimize total production costs, the regulator minimizesZ �+����

Z �+����

��1

2+

�2 � �12t (1� 2a�)

��1 +

�1

2� �2 � �12t (1� 2a�)

��2

�14�2d�1d�2

= �� �32

3pt:

Since this function is strictly decreasing in �, the regulator chooses the lowest possible level oftransparency. �

Proof of Proposition 2. The regulator chooses � to maximize

V (�) �Z �+����

Z �+����

W (:) 14�2d�1d�2

= �v �Z �+����

Z �+����

"Z 12+

�2��12t(1�2a�)

0

�(2�� 1)p� (�1) + �t (x� a�)2 + (1� �)�1

�dx+

�Z 112+

�2��12t(1�2a�)

�(2�� 1)p� (�2) + �t (1� a� � x)2 + (1� �)�2

�dx

#14�2d�1d�2

= �v � ��+ � � �t12 �p�t�2112�� 1

�+ ��

�16

q�t �

94

�: (A.5)

The rst-order necessary condition for an internal maximum is

@V (�)

@�= 0 , 2��t + (8� 18�)

q�t + 4� 7� = 0; (A.6)

which has a unique positive solution

� (�; t) = t4 �t4�2

�� (152� 175�)� 32 + (18�� 8)

p95�2 � 80�+ 16

�:

Evaluating the second-order su¢ cient condition at � (�; t),

@2V (�)

@�2

�����=�(�;t)

< 0 , 7�� 4 + 2��(�;t)t < 0 , � < � � 0:571:

Hence, for � < �� the function V (�) has a unique maximum at min f�; � (�; t)g, and � (�; t) � �for � � � � 0:516. Notice that, since by Lemma 3 social welfare is always higher at � than at� = 0, social welfare is also higher at min f�; � (�; t)g than at � = 0, when � � �. Moreover,simple computations show that @�(�;t)@t > 0 and

@�(�;t)@� < 0.

Consider now � > ��. The rst order condition (A.6) identies a unique local minimum of

V (�) since @2V (�)@�2

����=�(�;t)

> 0. Hence, the regulators program has a corner solution i.e., he

17

chooses either � or �. Using equation (A.5),

V (0)� V�t4

�= t

�1712��

34

�:

This di¤erence is strictly positive for � > ��. Since, by assumption, � is arbitrarily close to 0and � is arbitrarily close to t4 , continuity of V (�) and Lemma 3 imply that the social welfareis maximized at �, when � > ��. Retailerslocations are obtained by substituting the optimal �in a� (�). �

Proof of Lemma 5. First suppose that � � ��. In the proof of Proposition 2 we showed thatV (�) is strictly concave. By Lemma 3, V (�) > W (:)j�=0. Hence, we only need to prove thatV (�) > W (:)j�=0. This is true since, for � � 12 ,

V (�)� W (:)j�=0 = t12 (7�� 3) > 0:

Suppose now that � > ��. By Proposition 2, V (:) is minimized at � (�; t). Simple computationsshow that

V (� (�; t))� W (:)j�=0 =

= 2572 +8�2� 56� � 97�+

(9��4)28�2

p95�2 � 80�+ 16

�64�+p95�2�80�+16(9��4)�67�2�16

12�2

r8 + 44�2 � 38�� (9��4)

p95�2�80�+162 :

This is positive for � > ��. �

Two-part tari¤s. We show that, with secret contracts, RPM contracts are equivalent to two-part tari¤ contracts composed by a wholesale price and a franchise fee. In this framework, Mio¤ers a menu (wi(�i); Ti(�i))�i2� to Ri, where wi(�i) is the wholesale price paid by Ri for eachunit of the good produced by Mi and Ti(�i) is the franchise fee, given Ris report.

The timing of the game is modied as follows:

1. Retailers simultaneously and independently choose their locations: a1 and a2.

2. Retailers privately observe their costs.

3. Manufacturers o¤er contracts and retailers choose whether to accept them. If Ri acceptsMis o¤er, then he reports his type and pays the wholesale price and the franchise fee.

4. Retailers choose retail prices and the market clears.

Suppose that manufacturers and retailers are vertically separated. We focus on a symmetricseparating equilibrium. Given a report �̂i and the locations chosen by retailers, Ri chooses pi tosolve

maxpi(pi � wi(�̂i)� �i)

Z �+����

Di(pi; ~pj (�j))dF (�j) ;

18

where ~pj (�j) is the price that Ri expects Rj to choose in equilibrium as a function of his type.This yields the rst-order conditionsZ �+�

���Di(~pi(wi(�̂i); �i); ~pj (�j))dF (�j)�

~pi(wi(�̂i); �i)� wi(�̂i)� �i2t(1� ai � aj)

= 0

) ~pi(�i; wi(�̂i)) =�i +

R �+���� ~pj (�j) dF (�i) + wi(�̂i)

2+

+ t2 (1� ai � aj) (1 + ai � aj) ; i; j = 1; 2: (A.7)

Taking expectations for �̂i = �i,Z �+����

~pi(�i)dF (�i) =�+

R �+���� ~pj (�j) dF (�j) +

R �+���� wi (�i) dF (�i)

2+

+ t2 (1� ai � aj) (1 + ai � aj) ; i; j = 1; 2:

Solving the systemZ �+����

~pi(�i)dF (�i) = �+23

Z �+����

wi (�i) dF (�i) +13

Z �+����

wj (�j) dF (�j)+

+ t3 (3 + ai � aj) (1� ai � aj) ; i; j = 1; 2:

Hence, substituting in (A.7),

~pi(wi(�̂i); �i) =12

��+ �i + wi(�̂i)

�+ 16

Z �+����

wi (�i) dF (�i)+

+13

Z �+����

wj (�j) dF (�j) +t3 (3 + ai � aj) (1� ai � aj) :

In a separating equilibrium, the contract o¤ered by Mi must be incentive feasible i.e., itmust satisfy Ris participation and incentive compatibility constraints. Ris information rent is

Ui (�i) = (~pi(wi(�i); �i)� wi(�i)� �i)Z �+����

Di(~pi(wi(�i); �i); ~pj (�j))dF (�j)� Ti(�i):

Incentive compatibility implies

Ui(�i) = max�̂i2�

�(~pi(wi(�̂i); �i)� wi(�̂i)� �i)

Z �+����

Di(~pi(wi(�̂i); �i); ~pj (�j))dF (�j)� Ti(�̂i)�;

which yields the rst- and second-order local conditions for incentive compatibility

_U (�i) = �Z �+����

Di(~pi(wi(�i); �i); ~pj (�j))dF (�j) : (A.8)

19

and

��@~pi(wi(�i); �i)

@�i+@~pi(wi(�i); �i)

@wi

� Z �+����

@@~piDi(~pi(wi(�i); �i); ~pj (�j))dF (�j) � 0

) @~pi(wi(�i); �i)@�i

+@~pi(wi(�i); �i)

@wi� 0: (A.9)

Conditions (A.8) and (A.9), together with the participation constraint

Ui (�i) � 0; 8�i 2 �; (A.10)

dene the set of incentive feasible allocations for Mi and Ri:Mi maximizesZ �+����

Z �+����

�Di(~pi(wi(�i); �i); ~pj (�j)) (~pi(wi(�i); �i)� wi(�i)� �i)� Ui (�i)

dF (�j)dF (�i) ;

subject to conditions (A.8), (A.9) and (A.10). We rst assume, and then check ex post, that(A.10) is satised. Then Ui (�i) is decreasing and the participation constraint binds when �i =�+ �. Hence,

Ui (�i) =

Z �+��i

Z �+����

Di(~pi(wi(x); x); ~pj (�j))dxdF (�j): (A.11)

Using expression (A.11) integrating by parts and substituting intoMis objective function, yieldthe relaxed program

maxwi(:)

Z �+����

Z �+����

�Di(~pi(wi(�i); �i); ~pj (�j))

�pi(�i)� wi(�i)� �i �

F (�i)

f (�i)

��dF (�j)dF (�i) :

The rst-order necessary and su¢ cient condition for Mis program is

@~pi(wi(�i); �i)

@wi(�i)

Z �+����

Di(~pi(wi(�i); �i); ~pj (�j))dF (�j)+

� 12t (1� ai � aj)

�~pi(wi(�i); �i)� �i �

F (�i)

f (�i)

�@~pi(wi(�i); �i)

@wi(�i)= 0:

Simplifying,Z �+����

Di(~pi(wi(�i); �i); ~pj (�j))dF (�j)�1

2t (1� ai � aj)

�~pi(wi(�i); �i)� �i �

F (�i)

f (�i)

�= 0:

This is exactly the same rst-order condition of the maximization problem with RPM. By settingwi(�i) =

F (�i)f(�i)

, Ri chooses a retail price ~pi(wi(�i); �i) = p�i (�i), as dened in Lemma 2, and thelocations with a two-part tari¤ contract are the same as the ones with RPM.

20

References

[1] Bennardo, A., M. Pagano and S. Piccolo (2010), Multiple-Bank Lending, CreditorRights and Information Sharing.CSEF Working Paper n. 211.

[2] Blair, F., and T. Lewis (1994), Optimal Retail Contracts with Asymmetric Informationand Moral Hazard.RAND Journal of Economics, 25, 284-296.

[3] Calzolari, G., and A. Pavan (2006), On the Optimality of Privacy in Sequential Con-tracting.Journal of Economic Theory, 130, 168-204.

[4] DAspremont, C., J. Gabszewicz and J.-F. Thisse (1979), On Hotellings Stabilityin Competition.Econometrica, 17, 1145-1151.

[5] De Palma, A., V. Ginsburgh, Y. Papageorgiou and J.-F. Thisse (1985), The Prin-ciple of Minimum Di¤erentiation Holds Under Su¢ cient Heterogeneity.Econometrica,53, 767-782.

[6] Economides, N. (1986), Minimal and Maximal Product Di¤erentiation in HotellingsDuopoly.Economic Letters, 21, 67-71.

[7] Hotelling, H. (1929), Stability in Competition.Economic Journal, 39, 41-57.

[8] Gal-Or, E. (1991), Vertical Restraints with Incomplete Information.Journal of Indus-trial Economics, 39, 503-516.

[9] Gal-Or, E. (1999), Vertical Integration or Separation of the Sales Functions as Impliedby Competitive Forces.International Journal of Industrial Organization, 17, 641-662.

[10] Maier, N., and M. Ottaviani (2009), Information Sharing in Common Agency: Whenis Transparency Good?Journal of the European Economic Association, 7, 162-187.

[11] Kuhn, K.-U. (1997), Nonlinear Pricing in Vertically Related Duopolies.RAND Journalof Economics, 28, 37-62.

[12] Lafontaine, F., and M. Slade (1997), Retailing Contracting: Theory and Practice.Journal of Industrial Economics, 45, 1-25.

[13] Lanfontaine, F., and K. L. Shaw (1999), The Dynamics of Franchise Contracting:Evidence from Panel Data.Journal of Political Economy, 107, 1041-1080.

[14] Martimort, D. (1996), Exclusive Dealing, Common Agency, and Multiprincipals Incen-tive Theory.RAND Journal of Economics, 27, 1-31.

[15] Martimort, D., and S. Piccolo (2007), Resale Price Maintenance under AsymmetricInformation.International Journal of Industrial Organization, 25, 315-339.

[16] Matsushima, N. and T. Matsumura (2003), Cost Uncertainty and Partial Agglomera-tion.Mimeo, Tokio University.

[17] Pagnozzi, M., and S. Piccolo (2010), Vertical Separation with Private Contracts.Economic Journal, forthcoming.

[18] Taylor, C. (2004), Consumer Privacy and the Market for Customer Information.RANDJournal of Economics, 35, 63151.

[19] Tirole, J. (1988), The Theory of Industrial Organization. Boston, MIT press.

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